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### Chebyshev differentiation matrix

We conclude with a discussion on Chebyshev differencing. Starting with grid values at Chebyshev points , one constructs the Chebyshev interpolant

One can compute , efficiently via the cos-FFT with operations. Next, we differentiate in Chebyshev space

In this case, however, is not an eigenfunction of ; instead - being a polynomial of degree , can be expressed as a linear combination of (in fact is even/odd for even/odd k's): with we obtain

and hence

Rearranging we get (here, indicates halving the last term)

and similarly for the second derivative

The amount of work to carry out the differentiation in this form is operations which destroys the efficiency. Instead, we can employ the recursion relation which follows directly from (app_cheb.44)

To see this in a different way we note that

and hence

as asserted. In general we have

With this, can be evaluated using operations, and the differentiated polynomial at the grid points is computed using another cos-FFT employing operations

with spectral/exponential error

The matrix representation of Chebyshev differentiation, , takes the almost antisymmetric form (here except for )